To solve for the new weighted average when the ratio changes, we start by understanding how a weighted average is calculated.
Given two points \( a = 3 \) and \( b = 12 \), and a ratio of 2, we assume there are \( 2x \) points at 3 and \( x \) points at 12. The total number of points is \( 2x + x = 3x \).
The weighted average \( A \) is calculated as follows:
\[ A = \frac{(2x \cdot 3) + (x \cdot 12)}{3x} \]
Calculating the numerator:
\[ (2x \cdot 3) + (x \cdot 12) = 6x + 12x = 18x \]
Now substituting back to find the average:
\[ A = \frac{18x}{3x} = 6 \]
This confirms the original weighted average is \( 6 \).
Next, we change the ratio to 1, meaning we now have an equal number of points at 3 and 12. We set \( x \) points at 3 and \( x \) points at 12.
The new weighted average \( A \) now becomes:
\[ A = \frac{(x \cdot 3) + (x \cdot 12)}{x + x} \]
Calculating the numerator again:
\[ (x \cdot 3) + (x \cdot 12) = 3x + 12x = 15x \]
Now substituting back to find the new average:
\[ A = \frac{15x}{2x} = \frac{15}{2} = 7.5 \]
Therefore, the new weighted average when the ratio changes to 1 is \( \boxed{7.5} \).
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